# Guided-Wave Electro-Optic Modulators

### Share this post

This is a continuation from the previous tutorial - electro-optic modulators.

Optical waveguides possess many unique characteristics that do not exist in bulk optics. An important one is their ability to guide optical waves within a small cross-sectional area over a long distance. This allows for the possibility of using the transverse modulation scheme to realize very efficient modulators at very low modulation voltage. In bulk optics, the ratio of the length to the cross-sectional dimensions is limited by the diffraction effect, limiting the advantage that can be realized using transverse modulation. This limitation does not exist in waveguide optics. Another unique characteristic is the existence of waveguide modes. This results in many phenomena that have no counterpart in bulk optics, such as mode coupling, mode conversion, and modal dispersion. These unique features are the basis of many devices that take advantage of the waveguide configuration. In addition, guided-wave electro-optic devices are important building-block components of integrated optical and integrated optoelectronic systems.

The modulation electric field in a waveguide is usually the fringe field around surface electrodes or, in some cases of semiconductor waveguides, the field resulting from a junction voltage drop. Figure 6-6 shows the two commonly used approaches for buried waveguides, particularly the Ti-diffused LiNbO_{3} waveguides, using surface-loading electrodes.

In the configuration shown in figure 6-6(a), the electrodes are placed on two sides of the waveguide, and the horizontal electric field \(E_{0\parallel}\) is applied. In the configuration shown in figure 6-6(b), one of the electrodes is placed directly over the waveguide, and the applied electric field is the vertical \(E_{0\perp}\). The buried waveguide shown in figure 6-6 is a channel waveguide, but the index step at the air-crystal interface along the vertical direction is much higher than those at other waveguide boundaries. Therefore, modes with electric fields polarized mainly parallel to the air-crystal interface are called ** TE-like modes**, whereas those with electric fields polarized mainly perpendicular to this interface are called

**. When an electrode is placed directly over a waveguide, an insulating buffer layer, usually SiO**

*TM-like modes*_{2}or Al

_{2}O

_{3}, between the electrode and the substrate crystal is needed to ensure low loss for TM-like modes, as also shown in figure 6-6(b).

In a waveguide, the modulation electric field applied to a particular waveguide mode depends on a number of parameters, including the geometric dimensions of the waveguide structure and the optical field distribution of the mode. In general, the modulation field is not uniformly distributed across the mode field distribution. The effect of electro-optic modulation in a waveguide can be calculated using the coupled-mode theory discussed in the coupled-mode theory tutorial.

For modulation on a single mode, the effect is to introduce a change in the propagation constant of the mode. This change is equal to the self-coupling coefficient of the mode given by

\[\tag{6-68}\Delta\beta_\nu=\kappa_{\nu\nu}=\omega\int\limits_{-\infty}^{\infty}\hat{\boldsymbol{\mathcal{E}}}^*_\nu\cdot\Delta\boldsymbol{\epsilon}\cdot\hat{\boldsymbol{\mathcal{E}}}_\nu\text{d}\boldsymbol{\rho}\]

where \(\Delta\boldsymbol{\epsilon}\) is the electro-optically induced change in the dielectric permittivity tensor and \(\boldsymbol{\rho}\) is the two-dimensional vector in the cross-sectional plane of the waveguide.

As an example, we consider phase modulation of a TE-like mode in a waveguide modulator that is fabricated in a LiNbO_{3} crystal with the crystal surface perpendicular to the \(x\) principal axis and the longitudinal direction of the waveguide parallel to the \(y\) principal axis. This arrangement is shown in figure 6-7(a) and is referred to as ** \(y\) propagating** in an

**crystal.**

*\(x\)-cut*The modulation field appearing in the waveguide area is \(E_{0\parallel}\), which is \(E_{0z}\) in this configuration. Because a TE-like mode of this waveguide is predominantly polarized in the \(z\) direction and \(\Delta\epsilon_{zz}=-n_e^4r_{33}E_{0z}\) from (6-24) [refer to the Pockels effect tutorial], we have

\[\tag{6-69}\begin{align}\Delta\beta_\text{TE}&=-n_e^4r_{33}\omega\epsilon_0\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}E_{0z}(x,z)|\hat{\boldsymbol{\mathcal{E}}}_\text{TE}(x,z)|^2\text{d}x\text{d}z\\&=-\frac{n_e^4r_{33}}{2}\frac{V}{s_e}\frac{\omega^2\mu_0\epsilon_0}{\beta_\text{TE}}\Gamma_\text{TE}\\&=-\frac{\pi}{\lambda}n_e^3r_{33}\frac{V}{s_e}\Gamma_\text{TE}\end{align}\]

where \(V\) is the applied voltage, \(s_e\) is the separation between the electrodes, \(\beta_\text{TE}\) is approximated by \(n_e\omega/c\), (44) from the wave equations for optical waveguides tutorial is used to normalize the mode filed, and

\[\tag{6-70}\begin{align}\Gamma_\text{TE}&=\frac{s_e}{V}\frac{2\beta_\text{TE}}{\omega\mu_0}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}E_{0z}(x,z)|\hat{\boldsymbol{\mathcal{E}}}_\text{TE}(x,z)|^2\text{d}x\text{d}z\\&=\frac{s_e}{V}\frac{\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}E_{0z}(x,z)|\hat{\boldsymbol{\mathcal{E}}}_\text{TE}(x,z)|^2\text{d}x\text{d}z}{\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}|\hat{\boldsymbol{\mathcal{E}}}_\text{TE}(x,z)|^2\text{d}x\text{d}z}\end{align}\]

is the ** overlap factor**, which accounts for the overlap between the modulation electric field and the optical mode. The overlap factor has a value between 0 and 1. The total electro-optically induced phase shift of this mode over length \(l\) of the modulator is simply \(\Delta\varphi=\Delta\beta_\text{TE}l\).

Comparing the result obtained above with that in (6-41) [refer to the electro-optic modulators tutorial] for the bulk phase modulator, we find that the net effect for a waveguide mode \(\nu\) can be approximated by using a single uniform effective modulation electric field given by

\[\tag{6-71}E_\text{eff}=\Gamma_\nu\frac{V}{s_e}\]

where \(\Gamma_\nu\) is evaluated using the appropriate modulation field component for the device configuration under consideration. For example, \(E_{0\parallel}\) is used for the electrode configuration in figure 6-6(a), while \(E_{0\perp}\) is used for the configuration in figure 6-6(b). The value of \(\Gamma_\nu\) depends on the electrode configuration and is different for different waveguide modes in the same structure. For a given configuration and a given waveguide mode, it increases monotonically as the ratio of the electrode separation to the horizontal waveguide width increases.

**Example 6-5**

An \(x\)-cut, \(y\)-propagating LiNbO_{3} single-mode waveguide phase modulator for \(\lambda\) = 1.3 μm, as shown in figure 6-7(a), has a gap separation of \(s_e\) = 20 μm between its electrodes and an overlap factor of \(\Gamma_\text{TE}\) = 0.57 for its TE-like mode. It is modulated with an applied voltage of \(V\) = 12 V. What is the effective modulation electric field strength? Find the electro-optically induced change in the propagation constant of the TE-like mode. If an electro-optically controlled phase shift of \(\pi\) is desired, what is the required length of the device?

According to (6-71), the effective modulation electric field is

\[E_\text{eff}=\Gamma_\text{TE}\frac{V}{s_e}=0.57\times\frac{12}{20\times10^{-6}}\text{Vm}^{-1}=342\text{ kVm}^{-1}\]

We know that \(n_e\) = 2.145 for LiNbO_{3} at \(\lambda\) = 1.3 μm. We then find, using (6-69), that

\[\begin{align}\Delta\beta_\text{TE}&=-\frac{\pi}{1.3\times10^{-6}}\times2.145^3\times30.8\times10^{-12}\times\frac{12}{20\times10^{-6}}\times0.57\text{ m}^{-1}\\&=-251.23\text{ m}^{-1}\end{align}\]

For \(\Delta\varphi=\pi\), the required length of the device is

\[l=\frac{\pi}{|\Delta\beta_\text{TE}|}=12.5\text{ mm}\]

In the following, we discuss a few important electro-optic waveguide devices. The principle of phase modulation discussed in the preceding section can be applied directly to guide-wave phase modulators. Amplitude modulation and switching functions using guide-wave devices are typically realized using either waveguide interferometers or directional couplers. Polarization modulation is accomplished through electro-optically controlled coupling and conversion between guided modes of different polarizations. Other functions, such as frequency filtering, are also possible using guide-wave devices. Preferably, single-mode waveguide devices are used to attain the best performance at the lowest modulation voltage.

**Mach-Zehnder waveguide interferometers**

Guided-wave electro-optic phase modulators can be used to construct waveguide interferometers for effective amplitude modulation of guided optical waves. A Mach-Zehnder waveguide interferometer consists of two parallel waveguides connected at the input and output ends, respectively, by beam-splitting and beam-combining optical couplers. These couplers can be \(Y\)-junction waveguides, as in the devices shown in figure 6-8 below, or directional couplers, as shown in figure 6-9 below.

Modulation electric fields are applied to two parallel waveguides, which form the two arms of the interferometer and are sufficiently separated to avoid direct coupling between them. Each waveguide by itself functions as an electro-optic phase modulator. Constructive or destructive interference occurs at the output coupler if the phase difference between the two arms is, respectively, an even or odd multiple of \(\pi\). By electro-optically controlling this phase difference through the applied voltage, the amplitude of the guided optical field at the output can be modulated.

The Mach-Zehnder waveguide interferometer shown in figure 6-8(a) uses Y-junction couplers and is fabricated in an \(x\)-cut, \(y\)-propagating LiNbO_{3} crystal. To use the largest electro-optic coefficient \(r_{33}\) in LiNbO_{3}, both the modulation electric field and the optical field have to be polarized in the \(z\) direction. This requirement can be fulfilled by using the electrodes shown in figure 6-8(a) for TE-like modes. In this electrode configuration, the modulation voltage is applied to the central electrode while the outer electrodes are grounded. The modulation electric fields appearing in the two arms point in opposite directions, resulting in a ** push-pull operation** with equal but opposite phase shifts int he optical waves propagating through the two arms. For an interferometer with identical arms, any other background phase shifts are exactly cancelled. Thus the total phase difference is twice the electro-optically induced phase shift in each arm. If the two arms are identical single-mode waveguides, the phase difference induced by a modulation voltage \(V\) for a TE-like mode is

\[\tag{6-72}\Delta\varphi=\frac{2\pi}{\lambda}n_e^3r_{33}\Gamma_\text{TE}\frac{l}{s_e}V=\pi\frac{V}{V_\pi}\]

where

\[\tag{6-73}V_\pi=\frac{\lambda}{2n_e^3r_{33}\Gamma_\text{TE}}\frac{s_e}{l}\]

is the half-wave voltage corresponding to a phase difference of \(\pi\) between the two arms. For a TM-like mode, we have

\[\tag{6-74}V_\pi=\frac{\lambda}{2n_o^3r_{13}\Gamma_\text{TM}}\frac{s_e}{l}\]

The half-wave voltage for a TM-like mode is more than three times that for a TE-like mode of a similar overlap factor. Therefore, this particular interferometer favors operation with a TE-like mode.

**Example 6-6**

A Mach-Zehnder waveguide interferometric modulator for \(\lambda\) = 1.3 μm using Y junctions as shown in figure 6-8(a) consists of two parallel \(x\)-cut, \(y\)-propagating LiNbO_{3} single-mode waveguide phase modulators with \(s_e\) = 20 μm in a push-pull configuration. Both waveguides have \(\Gamma_\text{TE}=\Gamma_\text{TM}=\) 0.57 and \(l\) = 12.5 mm, like the one discussed in example 6-5 above. Find the half-wave voltages for the TE-like and TM-like modes, respectively.

We find from (6-73) that \(V_\pi\) for the TE-like mode is

\[V_\pi=\frac{1.3\times10^{-6}}{2\times2.145^3\times30.8\times10^{-12}\times0.57}\times\frac{20\times10^{-6}}{12.5\times10^{-3}}\text{V}=6\text{V}\]

which is half of the value of \(V_\pi\) for the phase modulator in Example 6-5 because of the push-pull operation of the interferometer. We know that \(n_o\) = 2.222 for LiNbO_{3} at \(\lambda\) = 1.3 μm. Therefore, from (6-74), we find that \(V_\pi\) for the TM-like mode is

\[V_\pi=\frac{1.3\times10^{-6}}{2\times2.222^3\times8.6\times10^{-12}\times0.57}\times\frac{20\times10^{-6}}{12.5\times10^{-3}}\text{V}=19.3\text{V}\]

which is larger than \(V_\pi\) for the TE-like mode because \(r_{13}\lt{r_{33}}\) for LiNbO_{3}.

In comparison, figure 6-8(b) shows a Mach-Zehnder interferometer fabricated on a \(z\)-cut, \(x\)-propagating LiNbO_{3} substrate. In this configuration, the electrodes have to be placed directly over the waveguides in order to use \(r_{33}\). For a push-pull operation with equal but opposite phase shifts in the two arms of this interferometer, only two electrodes are needed with one receiving the modulation voltage and the other grounded, as illustrated in figure 6-8(b). This interferometer favors a TM-like mode, which has a lower half-wave voltage of

\[\tag{6-75}V_\pi=\frac{\lambda}{2n_e^3r_{33}\Gamma_\text{TM}}\frac{s_e}{l}\]

than the half-wave voltage of

\[\tag{6-76}V_\pi=\frac{\lambda}{2n_o^3r_{13}\Gamma_\text{TE}}\frac{s_e}{l}\]

for a TE-like mode of a similar overlap factor.

For a Mach-Zehnder waveguide interferometer using Y junctions, if both input and output Y junctions are ideal 3-dB couplers, the power transmittance for a specific guided mode is

\[\tag{6-77}T=\frac{P_\text{out}}{P_\text{in}}=\cos^2\frac{\Delta\varphi}{2}=\frac{1}{2}(1+\cos\Delta\varphi)\]

For applications as a small-signal amplitude modulator, the device can be operated with a fixed bias voltage of \(V_b=V_\pi/2\) or \(-V_\pi/2\) for linear response. For applications as an ON-OFF modulator, no bias is needed. The maximum transmittance in the ON state versus the minimum transmittance in the OFF state is defined as the ** extinction ratio**. It is usually measured in decibels:

\[\tag{6-78}\text{ER}=-10\log\frac{T_\text{min}}{T_\text{max}}\;\text{(dB)}\]

One important advantage of the waveguide interferometer is that a very high extinction ratio can be accomplished with single-mode structures at a low modulation voltage. The major source of incomplete extinction in the OFF state comes from the imbalance between the two arms due to small fabrication errors, but a single-mode waveguide interferometer is very tolerant of this small imbalance. In a multimode structure, however, different modes have different \(V_\pi\) because \(\Gamma\) is different for different modes. Because this variation results in different \(\Delta\varphi\) for different modes at a specific modulation voltage, a high extinction ratio is difficult to accomplish when more than one mode is excited.

Instead of Y junctions, 3-dB directional couplers can be used at both input and output ends of a Mach-Zehnder waveguide interferometer. This type of Mach-Zehnder interferometer is called the ** balanced-bridge interferometer**. Figure 6-9 shows two examples. The phase-shifter section consists of two decoupled identical phase modulators. It has exactly the same function as that in a Mach-Zehnder interferometer using Y junctions. However, both input and output ports now have two channels. If straight waveguides are used, as in the example shown in figure 6-9(b), the waveguides have to be closely spaced to allow coupling in the coupler sections. Crosstalk due to coupling in the phase-shifter section has to be eliminated. This objective can be accomplished by etching a slot in the gap, as shown in figure 6-9(b), or by mismatching the two waveguides.

With input to only one channel, the straight transmission through the same channel at the output is

\[\tag{6-79}T=\sin^2\frac{\Delta\varphi}{2}=\frac{1}{2}(1-\cos\Delta\varphi)\]

where \(\Delta\varphi\) is the electro-optically induced phase difference between the two arms of the interferometer discussed above. The crossover efficiency to the other output channel is

\[\tag{6-80}\eta=1-T=\cos^2\frac{\Delta\varphi}{2}=\frac{1}{2}(1+\cos\Delta\varphi)\]

When used as a modulator, a balanced-bridge interferometer has two complementary output channels. Otherwise, it has similar characteristics to those of an interferometer using Y junctions. In addition, a balanced-bridge interferometer can also be used as an optical switch. When \(\Delta\varphi\) is 0 or any integral multiple of \(2\pi\), the interferometer is in the cross state because \(\eta=1\). This feature is expected because in this situation the phase-shifter section has null net effect, and the function of the interferometer is simply that of two serially connected 3-dB directional couplers. When \(\Delta\varphi\) is equal to any odd integral multiple of \(\pi\), the interferometer is in the parallel state with \(T=1\). By switching the control voltage for \(\Delta\varphi\) to have even or odd multiples of \(\pi\), the device can be electrically switched between the two switch states.

**Directional coupler switches**

A very important practical application of directional couplers is to use t hem as optical switches, which can be switched between cross and parallel states. This function requires switching the coupling efficiency \(\eta\) between the values of 1 and 0. For a fabricated device with fixed geometric parameters, this can be done by varying the phase mismatch or the coupling coefficient between the two waveguides through electro-optically induced changes in the refractive index of the waveguide material. For simplicity, we consider in the following only symmetric directional couplers where the two waveguide channels have identical geometric and material parameters. Any differences between the two channels are thus induced solely by the applied electric field through the electro-optic effect. Similar concepts can be applied to asymmetric directional couplers as well.

Figure 6-10(a) shows the basic structure of an electro-optic directional coupler switch. This structure has a two-electrode configuration similar to that of the two-electrode Mach-Zehnder interferometers. The only difference is that the two waveguides in a directional coupler are coupled while those in an interferometer are isolated. If the device is fabricated on a \(z\)-cut, \(x\)-propagating LiNbO_{3} substrate, the effect of the applied voltage is also to induce a phase difference of \(\Delta\varphi=\pi V/V_\pi\), with \(V_\pi\) given by (6-75) and (6-76) for TM-like and TE-like modes, respectively.

Without an applied voltage, the symmetric coupler is perfectly phase matched with \(\beta_a=\beta_b,\kappa_{aa}=\kappa_{bb}\), and \(\kappa_{ab}=\kappa_{ba}^*\equiv\kappa\), where \(\kappa\) is real and positive. The effective propagation constant for each waveguide is \(\beta=\beta_a+\kappa_{aa}=\beta_b+\kappa_{bb}\), and the phase-matched coupling length is \(l_c^\text{PM}=\pi/2\kappa\) given in (54) [refer to the directional couplers tutorial]. The phase difference \(\Delta\varphi\) induced by the applied voltage results in a phase mismatch of

\[\tag{6-81}2\delta=\Delta\beta=\frac{\Delta\varphi}{l}\]

between the two waveguides of the coupler. To first order, the changes in the coupling coefficient \(\kappa\) induced by the electric field can be neglected in this two-electrode configuration. More significant changes in \(\kappa\) are possible using a three-electrode configuration.

With a phase mismatch given by (6-81), the coupling efficiency of this switch is that given by (88) [refer to the two-mode coupling tutorial]

\[\tag{6-82}\eta=\frac{1}{1+\delta^2/\kappa^2}\sin^2\left(\kappa{l}\sqrt{1+\delta^2/\kappa^2}\right)\]

Because \(\Delta\varphi\) and hence the value of \(\delta\) are linearly proportional to the applied voltage, the switching function of the device can be accomplished by electrically switching \(\delta\) between the values corresponding to \(\eta=1\) and 0. However, it can be seen from (6-82) that if \(\delta\ne0\), it is not possible to have \(\eta=1\) though \(\eta=0\) is possible. Therefore, to allow access to both cross and parallel states, it is necessary to design the device to be in the cross state when there is no applied voltage, \(V=0\) and thus \(\delta=0\). This requirement means that the coupler has to be perfectly symmetric and its length has to be exactly one of the odd integral multiples of \(l_c^\text{PM}\):

\[\tag{6-83}l=(2n+1)l_c^\text{PM},\qquad{n=0,1,2,\ldots}\]

The parallel state can then be reached with an applied voltage to induce a \(\delta\) that satisfies \(\kappa{l}\sqrt{1+\delta^2/\kappa^2}=m\pi\) so that \(\eta=0\). Using (6-81) and (6-83), this condition can be cast in the following form:

\[\tag{6-84}\left(\frac{l}{l_c^\text{PM}}\right)^2+\left(\frac{\Delta\beta{l}}{\pi}\right)^2=4m^2,\qquad{m=0,1,2,\ldots}\]

These conditions for the cross and the parallel state are plotted in figure 6-10(b). As an example, we see that if \(l=l_c^\text{PM}\), the parallel state can be first reached with \(\Delta\varphi=\Delta\beta{l}=\sqrt{3}\pi\). This phase shift corresponds to a switching voltage \(V_s=\sqrt{3}V_\pi\), which is \(\sqrt{3}\) times that needed for the balanced-bridge interferometer switch discussed above to reach the first parallel state. As can be seen from figure 6-10(b), for \(l=l_c^\text{PM}\), the parallel state can be further reached with \(\Delta\beta{l}=\sqrt{4m^2-1}\pi\) at higher applied voltages, but the cross state exists only when \(\Delta\beta{l}=0\).

**Example 6-7**

A uniform-\(\Delta\beta\) directional coupler switch as shown in figure 6-10 for \(\lambda\) = 1.3 μm is fabricated on a \(z\)-cut, \(x\)-propagating LiNbO_{3} substrate. The gap separation between the electrodes is \(s_e\) = 5 μm, and the overlap factor for the TM-like mode is found to be \(\Gamma_\text{TM}\) = 0.247 for optical waveguides of 6 μm width. A desired coupling coefficient between the two parallel waveguides can be obtained by properly choosing the spacing between the parallel waveguides in the coupling section covered by the electrodes. If a switching voltage of 5 V for the TM-like mode is desired, what are the required coupling coefficient and the length of the coupling section?

The length of the coupling section can be chosen to be \(l=l_c^\text{PM}=\pi/2\kappa\), which is the shortest length required to have access to both cross and parallel states. Then the switching voltage is \(V_s=\sqrt{3}V_\pi\). For a switching voltage of 5 V, we find that \(V_\pi\) = 2.89 V. For this \(z\)-cut device in a push-pull configuration, \(V_\pi\) is that given by (6-75). Therefore, we find that the required length for \(V_\pi\) = 2.89 V is

\[l=\frac{\lambda}{2n_e^3r_{33}\Gamma_\text{TM}}\frac{s_e}{V_\pi}=\frac{1.3\times10^{-6}}{2\times2.145^3\times30.8\times10^{-12}\times0.247}\times\frac{5\times10^{-6}}{2.89}\text{m}=15\text{ mm}\]

The required coupling coefficient is

\[\kappa=\frac{\pi}{2l}=104.72\text{ m}^{-1}\]

The function of the directional coupler switch shown in figure 6-10 depends critically on the accuracy of fabrication because its cross state cannot be reached by tuning the applied voltage but requires precise symmetry and the exact length of the coupler. Any slight deviation in the symmetry or in the length results in crosstalk to other channel in the cross state. This limitation can be overcome by using the ** reversed-\(\Delta\beta\)** configuration shown in figure 6-11(a). In this configuration, voltages of equal magnitude but opposite polarities are applied to the split electrodes. If the electro-optically induced phase mismatch in the first section of a length of \(l/2\) is \(\Delta\beta=2\delta\), that in the second section, also of a length of \(l/2\), is \(-\Delta\beta=-2\delta\). By considering the device as two couplers in tandem and by solving the coupled-mode equations, it can be shown that the total coupling efficiency of this device is

\[\tag{6-85}\eta=\frac{4\kappa^2}{\beta_c^2}\sin^2\frac{\beta_cl}{2}\left(1-\frac{\kappa^2}{\beta_c^2}\sin^2\frac{\beta_cl}{2}\right)\]

where \(\beta_c=(\kappa^2+\delta^2)^{1/2}\), as defined in (61) [refer to the two-mode coupling tutorial]. By expressing (6-85) in terms of \(l/l_c^\text{PM}\) and \(\Delta\beta{l}/\pi\), the conditions for the parallel and the cross state, which have \(\eta=0\) and \(\eta=1\), respectively, can be plotted in the switching diagram shown in figure 6-11(b).

It is now possible to reach both the parallel and the cross state by controlling the applied voltage if \(l/l_c^\text{PM}\) is chosen to be within a proper range such as \(1\le{l/l_c^\text{PM}}\le3\). As an example, consider \(l=\sqrt{2}l_c^\text{PM}\). In this case, the cross state with \(\eta=1\) is reached when \(\Delta\beta=2\kappa\), thus \(\Delta\beta{l}=\sqrt{2}\pi\), and the parallel state with \(\eta=0\) is reached when \(\Delta\beta=2\sqrt{7}\kappa\), thus \(\Delta\beta{l}=\sqrt{14}\pi\). This example is also illustrated in figure 6-11(b).

The possibility of reaching the cross state in this condition can be understood intuitively with the illustration shown in figure 6-12 for the conditions of \(l=\sqrt{2}l_c^\text{PM}\) and \(\Delta\beta=2\kappa\).

For a coupler with a uniform \(\Delta\beta\) across the two sections, we find from (6-82) that \(\eta=0\), and the device is in the parallel state rather than in the cross state. We also find by substituting \(l\) in (6-82) with \(l/2\) that at the end of the first section, the input power from one channel is equally divided between the two channels. Therefore, each section alone functions as a 3-dB coupler. Under the conditions considered here, the second section of the uniform-\(\Delta\beta\) coupler couples all of the power back to the original channel, resulting in \(\eta=0\) and the parallel state, as shown in figure 6-12(a). However, if the sign of phase mismatch is reversed in the second section, as is the case in the reversed-\(\Delta\beta\) coupler, the process in the second section is also reversed, resulting in \(\eta=1\) and the cross state. This process is shown in figure 6-12(b).

**Waveguide polarization modulators**

The function of a waveguide polarization modulator depends on the coupling between modes of different polarizations. As is true for any coupled-mode device, the key parameters to be considered are also the phase mismatch and the coupling coefficient between the two polarization modes.

Unlike the situation in the directional couplers discussed above, however, the two polarization modes are generally phase mismatched and are originally uncoupled. Therefore, the task of the modulation electric field is to create a sufficiently strong field-dependent coupling coefficient while simultaneously reducing or eliminating the phase mismatch if necessary.

For two orthogonally polarized modes to couple to one another, it is necessary to induce the corresponding off-diagonal elements in the dielectric permittivity tensor. For instance, if a LiNbO_{3} waveguide is fabricated with its structural axes lined up with the principal axes of the crystal, a modulation electric field \(E_{0y}\) will couple \(y\)- and \(z\)-polarized mode fields, as can be seen from (6-27) [refer to the Pockels effect tutorial], whereas an \(E_{0x}\) creates coupling between \(x\)- and \(y\)-polarized modes and that between \(x\)- and \(z\)-polarized modes, as can be seen from (6-32) [refer to the Pockels effect tutorial]. In contrast, an \(E_{0z}\) cannot create such coupling between any two orthogonal polarizations, as can be seen from (6-24) [refer to the Pockels effect tutorial].

As an example, we consider the coupling between fundamental TE-like and TM-like modes in an \(x\)-cut, \(y\)-propagating LiNbO_{3} waveguide shown in figure 6-13. Because the birefringence of LiNbO_{3} is relatively large, phase mismatch between the TE-like and the TM-like modes is contributed primarily by the fact that they have quite different indices, approximately \(n_e\) and \(n_o\), respectively:

\[\tag{6-86}\Delta\beta=\beta_\text{TM}-\beta_\text{TE}\approx\frac{2\pi}{\lambda}(n_o-n_e)\]

This large phase mismatch has to be compensated before any significant coupling between these two modes can take place. The required phase matching can be accomplished by using a grating of a proper period \(\Lambda\), as discussed in the grating waveguide couplers tutorial and indicated by (7) [refer to the grating waveguide couplers tutorial]. For perfect phase matching with \(\delta=0\), we need \(qK=-\Delta\beta\). Therefore, the grating period has to be one of the those given by

\[\tag{6-87}\Lambda=-q\frac{2\pi}{\Delta\beta}\approx-q\frac{\lambda}{n_o-n_e},\qquad{q=-1,-2,\ldots},\]

where \(q\) takes on negative integral values because \(n_o\gt{n_e}\) for LiNbO_{3} and the value of \(\Lambda\) has to be positive. For a first-order grating (\(q=-1\)), the phase-matching condition requires that \(\Lambda\approx\) 7, 12.5 and 18 μm for \(\lambda\) = 0.6, 1, and 1.3 μm, respectively. Such a grating can be generated by using the periodic electrode shown in figure 6-13 to create a periodic electro-optically modulated \(\Delta\boldsymbol{\epsilon}\) along the propagation direction of the waveguide. The coupling coefficient is then given by

\[\tag{6-88}\kappa_\text{EM}(q)=\kappa_\text{ME}^*(q)=\frac{\omega}{\Lambda}\displaystyle\int\limits_0^\Lambda\text{d}y\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}z\hat{\boldsymbol{\mathcal{E}}}^*_\text{TE}(x,z)\cdot\Delta\boldsymbol{\epsilon}(x,y,z)\cdot\hat{\boldsymbol{\mathcal{E}}}_\text{TM}(x,z)\text{e}^{-\text{i}qKy}\]

where \(\Delta\boldsymbol{\epsilon}\) is periodic in \(y\) for the device configuration shown in figure 6-13.

Since the TE-like and TM-like modes are mainly polarized in the \(z\) and \(x\) directions, respectively, they are coupled primarily through the off-diagonal \(zx\) and \(xz\) terms of \(\Delta\boldsymbol{\epsilon}\) in (6-32) [refer to the Pockels effect tutorial] that are contributed by the effect of the \(E_{0x}\) component of the periodic modulation electric field. Although a periodic \(E_{0y}\) component also exists, its contribution is not significant. For a first-order grating, this leads to the following coupling coefficient:

\[\tag{6-89}\begin{align}\kappa&=\kappa_\text{EM}(q=-1)=\kappa_\text{ME}^*(q=-1)\\&\approx\frac{\omega}{\Lambda}\displaystyle\int\limits_0^\Lambda\text{d}y\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}z\hat{\mathcal{E}}_\text{TE,z}^*(x,z)(-\epsilon_0n_o^2n_e^2r_{42})E_{0x}(x,y,z)\hat{\mathcal{E}}_\text{TM,x}(x,z)\text{e}^{\text{i}Ky}\\&\approx-\frac{\pi}{\lambda}n_o^{3/2}n_e^{3/2}r_{42}\Gamma_\text{EM}\frac{V}{s_e}\end{align}\]

where

\[\tag{6-90}\begin{align}\Gamma_\text{EM}&=\frac{2\beta_\text{TE}^{1/2}\beta_\text{TM}^{1/2}}{\omega\mu_0}\frac{s_e}{V}\frac{1}{\Lambda}\displaystyle\int\limits_0^\Lambda\text{d}y\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}zE_{0x}\hat{\mathcal{E}}^*_\text{TE,z}\hat{\mathcal{E}}_\text{TM,x}\text{e}^{\text{i}Ky}\\&\approx\frac{S_e}{V}\frac{\displaystyle\frac{1}{\Lambda}\int\limits_0^\Lambda\text{d}y\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}zE_{0x}\hat{\mathcal{E}}^*_\text{TE,z}\hat{\mathcal{E}}_\text{TM,x}\text{e}^{\text{i}Ky}}{\left(\displaystyle\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}z|\hat{\mathcal{E}}_\text{TE,z}|^2\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\text{d}x\text{d}z|\hat{\mathcal{E}}_\text{TM,x}|^2 \right)^{1/2}}\end{align}\]

is the overlap factor for electro-optic coupling of TE-like and TM-like modes. If perfect phase matching is accomplished by accurate selection of the grating period, the coupling efficiency is simply that given by (85): \(\eta_\text{PM}=\sin^2|\kappa|l\) [refer to the two-mode coupling tutorial]. Because \(|\kappa|\) can now be controlled by the modulation voltage, the coupling length \(l_c^\text{PM}=\pi/2|\kappa|\) can be varied by varying the voltage. For a given device of length \(l\), the coupling efficiency can then be varied between 0 and 1 through variation of the modulation voltage. In general, the device can be used to modulate or control the polarization of the guided optical wave, for example, to convert an elliptically polarized input beam into a linearly polarized output beam, or vice versa. When the voltage is set at a value for \(\eta_\text{PM}=1\), it is possible to convert a TE-like mode to a TM-like mode completely, or vice versa. In this instance, the device functions as a ** TE-TM mode converter**.

**Example 6-8**

An \(x\)-cut, \(y\)-propagating LiNbO_{3} waveguide TE-TM mode converter as shown in figure 6-13 for \(\lambda\) = 1.3 μm consists of a periodic interdigital electrode that has a period of \(\Lambda\) = 18 μm for phase matching. The inter-electrode gap is \(s_e\) = 4 μm, and the overlap factor for TE-TM electro-optic coupling is \(\Gamma_\text{EM}\) = 0.1. The total length of the electrode section is \(l\) = 10 mm. Find the mode-conversion voltage that is required for complete conversion between TE and TM modes.

The required electro-optically controlled coupling coefficient for complete TE-TM mode conversion is \(|\kappa|=\pi/2l\) for the device of length \(l\). By using (6-89), the mode-conversion voltage can be expressed as

\[\tag{6-91}V_\text{EM}=\frac{\lambda|\kappa|s_e}{\pi{n_o}^{3/2}n_e^{3/2}r_{42}\Gamma_\text{EM}}=\frac{\lambda}{2n_o^{3/2}n_e^{3/2}r_{42}\Gamma_\text{EM}}\frac{s_e}{l}\]

with the given parameters of the device, we find that

\[V_\text{EM}=\frac{1.3\times10^{-6}}{2\times2.222^{3/2}\times2.145^{3/2}\times28\times10^{-12}\times0.1}\times\frac{4\times10^{-6}}{10\times10^{-3}}\text{ V}=8.92\text{ V}\]

Phase matching for the polarization modulator shown in figure 6-13 cannot be adjusted electrically but has to be accomplished by accurate selection and fabrication of the electrode period. Figure 6-14 below shows a few other configurations of waveguide polarization modulators, which are fabricated on \(z\)-propagating LiNbO_{3} substrates.

In the configuration shown in figure 6-14(a), a \(y\)-cut substrate is used. A two-electrode structure provides the horizontal modulation field component \(E_{0x}\) for coupling between TE-like and TM-like modes, which are polarized mainly in the \(x\) and \(y\) directions, respectively. The two polarizations have the same ordinary index because the propagation direction is along the optical axis of the crystal. The only phase mismatch between the TE-like and the TM-like modes is caused by modal dispersion. Because modal dispersion is small in a weakly guiding waveguide, it need not be intentionally compensated if the coupling coefficient is made sufficiently large so that \(|\kappa|\gg\delta\). Therefore, a fixed bias voltage can be used to provide a large bias \(|\kappa|\) to reduce the effect of the phase mismatch effectively. A similar situation applies to polarization modulators fabricated in III-V semiconductors, as can be seen from (6-34) [refer to the Pockels effect tutorial]. In fact, because a III-V semiconductor is not birefringent, the waveguide can be fabricated along any direction, not necessarily parallel to a crystal axis, for modes with different polarizations to suffer only modal dispersion.

In \(z\)-propagating LiNbO_{3} waveguides, the slight modal dispersion can be further compensated, if desired, by applying an \(E_{0y}\) component in addition to \(E_{0x}\). This can be done using a three-electrode configuration with asymmetrically applied voltages, as shown in figure 6-14(b), or using an asymmetrically placed two-electrode configuration with one electrode directly on top of the waveguide, as shown in figure 6-14(c). Compensation of the phase mismatch can be accomplished by adjusting \(E_{0y}\) through the applied voltage because \(E_{0y}\) induces equal but opposite changes in the refractive indices along \(x\) and \(y\) directions, as can be seen from (6-27) [refer to the Pockels effect tutorial].

The next part continues with the traveling-wave modulators tutorial.